Alistair Windsor (Penn State)

Mixed spectrum reparameterizations of linear flow on ${\mathbb T}^2$.

Abstract. We consider time changes of an irrational flow on ${\mathbb T}^2$ defined by $\frac{dx}{dt} = \alpha \qquad \frac{dy}{dt} = 1$. The study of these time changes began with Kolmogorov and many of the basic questions in the area date from his 1954 I.C.M. address. The reparameterized flow remains minimal and uniquely ergodic but may exhibit other ergodic properties which are quite distinct from the original linear flow. That a plethora of ergodic properties can be obtained via continuous time changes follows from Kakutani equivalence. For sufficiently smooth reparameterizations the situation is more subtle and depends on the arithmetic properties of $\alpha$. For Liouville $\alpha$ the a generic $C^\infty$ time change results in a flow which is weak mixing. This is in stark contrast to the original linear flow which has pure point spectrum.

In joint work with B. Fayad [C.N.R.S.] and A. B. Katok we prove that for any Liouville flow there exist $C^\infty$ time changes for which the resulting flow has mixed spectrum.